2010/10/20 by Cinzia Casagrande, Casagrande, Cinzia · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1010.4129
openalex publication_date 2010/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the birational geometry of a Fano 4-fold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map f: X-->Y, where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if the difference of the Picard numbers of X and Y is 1. We first give a characterization of non-movable prime divisors in X, when X has Picard number at least 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, quasi-elementary. The main result is that the Picard number of X is at most 11 if X has an elementary rational contraction of fiber type, and 18 if X has a quasi-elementary rational contraction of fiber type.